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Identification of Patient Specific Parameters for a Minimal Cardiac Model
C. E. Hann1, J. G. Chase1, G. M. Shaw2, B. W. Smith3,
1
Bioengineering Centre, Department of Mechanical Engineering, University of Canterbury, Christchurch, New Zealand
2
Department of Intensive Care Medicine, Christchurch Hospital, Christchurch, New Zealand
3
Centre for Model-based Medical Decision Support, Aalborg University, Aalborg, Denmark
Abstract—A minimal1 cardiac model has been developed
which accurately captures the essential dynamics of the cardiovascular system (CVS). This paper develops an integral based
parameter identification method for fast and accurate
identification of patient specific parameters for this minimal
model. The integral method is implemented using a single
chamber model to prove the concept, and turns a previously
non-linear and non-convex optimization problem into a linear
and convex problem. The method can be readily extended to
the full minimal cardiac model and enables rapid identification
of model parameters to match a particular patient condition in
clinical real time (3-5 minutes). This information can then be
used to assist medical staff in understanding, diagnosis and
treatment selection.
Keywords—Cardiac model, Integral optimization, Patient
specific parameters, Convex, Diagnosis, Treatment selection.
I. INTRODUCTION
Optimising haemodynamics in the critically ill is an
important task for intensive care staff that must filter and
integrate a diverse range of information about a particular
patient’s circulation. Often this information is incomplete
and/or confusing leaving medical professionals to rely on
their experience and intuition, and often trial and error, to
identify and treat a patient’s condition. A “Minimal Model”
approach to CVS modeling means using a minimal number
of governing equations and parameters where other similar
models in the literature have been found to use many
variables and complex formula. A minimal model has been
developed by Smith et al [1], which is shown to adequately
provide appropriate magnitudes and trends in agreement
with existing data for a variety of physiologically verified
test cases simulating human CVS function [2].
To use this model to assist medical staff in diagnosis
and treatment there needs to be a method for identifying
patient specific parameters that is reasonably accurate and
fast. Ideally, the means of identifying the parameters should
be convex to avoid finding a false solution. A patient
specific model enables therapeutic choices to be tested and
can aid diagnosis of subtle haemodynamic behaviours.
The standard method for identifying parameters in a
physiological model described by differential equations is
non-linear regression [3]. This method also involves solving
the differential equations every time the parameters are
updated in the optimization. The problem with this standard
approach is that even if the differential equation is linear in
the parameters the numerical (or analytical) solution will, in
general, be non-linear in the parameters. Thus, the
optimization is non-linear and there is no guarantee of
finding the correct solution. In addition, these methods are
derivative based, amplifying the impact of noisy or
erroneous measurements.
In this paper, the optimization is formulated in terms of
integrals, which enables a set of linear equations in the
parameters to be set up. This approach has a unique linear
least squares solution so the optimization is convex and fast.
In addition, the differential equation is not required to be
solved, which significantly further reduces computation.
II. METHODOLOGY
For a single elastic chamber with inertia and constant
upstream and downstream pressures P1 and P3 the differential
equations are defined [1]:
V  Q1  Q 2
P  P2  Q1 R1
Q 1  1
L1
(1)
P  P3  Q 2 R2
Q 2  2
L2
where Q1 and Q2 are the flows in and out, L1 and L2 are
inertances of the blood, R1 and R2 are resistances and the
pressure in the chamber is defined:
P2  e(t ) E es (V  Vd )  (1  e(t )) P0 ( e  (V V )  1),
0
e(t )  e 80( t 0.375)
2
where E es is elastance, Vd is volume at zero pressure and P0 ,
 , and V0 define gradient, curvature and volume at zero
pressure of the EDPVR curve [1]. Fig. 1 summarises this
single chamber lumped parameter cardiac chamber model.
1
Funding for this research was provided by a New Zealand FRST PostDoctoral Fellowship
(2)
Fig. 1. The single cardiac chamber model.
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To demonstrate the integral approach P0 ,  ,V0 and Vd are
held constant and Ees , L1 , L2 , R1 and R2 are optimized to
match a given patient response. It is assumed that data is
available for the flows Q1 and Q2 .This data could be
obtained by echocardiography or by differentiating volume
data, which could be measured using ultrasound [4].
Consider the filling stage of the cardiac cycle where
Q2  0, Q 2  0, V  Q1
TABLE I
CONSTANTS USED IN SINGLE-CHAMBER SIMULATION
Description
Symbol
EDPVR volume
DSPVR volume
Constant
Heart rate
Constant
(3)
R1
R2
L1
L2
(4)
Pressure
t
(5)
0
Integrating (4) from t0 to t gives,
83000 N s m5
81000 N s m5
430000 N s2 m5
480000 N s2 m5
3 mmHG
P1
P3
The volume of the chamber is defined:
V (t )  V (t0 )  t Q1dt,
0 m3
33000 m-3
1.33 beats s-1
10 N m-2
P0
Inertance
L1Q 1  P1  P2  Q1 R1
0 m3


Resistance
and
Value
V0
Vd
100 mmHG
The driver function e(t ) is translated so that when
t  0 it coincides with the beginning of the filling stage.
The equations for V and Q then V and Q are solved
1
t
L1 (Q1 (t )  Q1 (t0 ))  P1t  P1t0  Ees  e(t )(V  Vd )dt
t0
t
t
t0
t0
  (1  e(t )) P0 (e (V V0 )  1)  R1  Q1dt
(6)
In the ejection stage, Q1  0, Q 1  0, V  Q2 and
L2Q 2  P2  P3  Q2 R2
(7)
2
numerically in MAPLE over one heart beat using a
procedure similar to that in [1]. Figs 1 and 2 show the
resulting PV and flow graphs.
The flows Q1 and Q2 are equally sampled at 21 points and 8
points, respectively, as the filling stage is longer than the
ejection stage. This sampling discretises the data. The data
is made continuous by cubic spline fitting as if it were
measured data. Equations (9) and (10) are then represented.
eqi(1)  L1Q1 (ti )  P1ti  Ees 0 e(t )(V  Vd )
ti
 t (1  e(t )) P0 (e (V V )  1)  R1 t Q1dt  0
which leads, after integration, to:
ti
ti
0
0
V (t )  V (t1 )  t Q2 dt,
(9)
0
t
eq (j 2 )  L2 Q 2 (t j )  P3 t j  P3 T1  E es T e(t )(V  V d )dt
tj
1
(8)
L2 (Q2 (t )  Q2 (t1 ))   P3t  P3t1  Ees t e(t )(V  Vd )dt
t
 t (1  e(t )) P0 ( e
t
 (V V0 )
0
(10)
1
 T (1  e(t )) P0 ( e  (V V )  1)  R 2 T Q 2 dt  0
tj
0
 1)dt  R2 t Q2 dt.
t
0
1
tj
1
0
For fixed values of t 0 , t1 and t , the integrals in (6) and (8)
are known constants since e(t ), V (t ), Vd , V0 , P0 ,  , Q1 and
Q 2 are given or measured. Also P1 and P3 are given so (6)
and (8) are linear in E es , L1 , L2 , R1 and R2 . By varying t 0 ,
t1 and t more linear equations in E es , L1 , L2 , R1 and R2 can
be set up. In the next section the problem is formulated so
that t0  0, V (0) is the minimum volume, V (t1 ) is the
maximum volume Q1 (0)  0 and Q2 (t1 )  0.
III. RESULTS
The constants P0 ,  , V0 , Vd , E es , L1 , L2 , R1 and R2 are
chosen using values from [1] as shown in Table 1.
where T1 is the time of ejection with 16 values of t i chosen
in the time interval when Q1  0 and 14 values of t j chosen in
the time interval when Q2  0 . The result is 30 equations
with 5 unknowns. These equations are solved by linear least
squares in MAPLE and the results are in Table 2.
TABLE 2: OPTIMISED PARAMETER VALUES
Parameter
True value
Optimised
value
Percentage error
E es
3.5555  108
3.5555  108
0.02
R1
R2
L1
L2
83000
81128
2.26
81000
81768
0.95
430000
430876
0.20
480000
479868
0.03
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The optimized values for the parameters are used to run the
model again just as in a clinical situation. Figs 1 and 2 are
the PV curve and flow comparisons with the results using
the true values, which show a very close match. Note that
the curves are essentially overlaid per the very small errors
in Table 2.
The relative percentage error is calculated for data
points equally spaced a distance 0.01 seconds apart from t =
0.01 to 0.43 seconds for the flow rate Q1 and t = 0.51 to 0.63
TABLE 3: MODEL RESPONSE ERROR WITH OPTIMISED VALUES
seconds for the flow rate Q 2 . The percentage error is also
calculated for data points a distance 0.01 seconds apart from
t = 0.01 to 0.74 for the PV curves. For the PV curve the
IV. DISCUSSION
percentage error
ei 
ei is calculated:
(V true (ti )  V optimised(t i )) 2  ( P2true (t i )  P2optimised(t i )) 2
(V true (t i )) 2  ( P2true (t i )) 2
(11)
Table 3 shows the error calculation results. The errors are
all less than 0.2 % further highlighting the results in Figs 1
and 2. The average error is less than 0.1 % validating this
parameter identification approach.
Fig. 1. PV curves for model with optimized values versus the model with
the true values.
Outflow
Mean
percentage
error
Standard
deviation
Q1
Q2
0.17
0.08
0.08
0.06
PV
0.09
0.06
The integral based optimization successfully identified the
patient specific parameters for the single chamber model.
This approach can be readily extended to the full model in
[1]. The use of integrals means any noise in the data will be
low pass filtered, and the optimization problem is made
linear and convex where current approaches are non-linear
and non-convex. In addition, the differential equation is not
required to be solved each time and initial conditions are not
needed. Thus, the issue of the incorrect initial conditions
leading to increased time for model convergence is avoided
[2].
These methods also result in a significant reduction in
computation. Note that the eV term in P2 could be
approximated by a piecewise linear function so that if  is
required to be optimized the equations will still be linear.
Extra parameters would be needed to represent eV but the
advantage of using integrals is that as many equations as
required can be set up by integrating along different time
intervals, only limited by the resolution and extent of the
data. The number of intervals can be chosen so that the
number of equations is greater than the number of variables
ensuring a unique solution.
The same idea can be applied to other non-linearities in
the full model. Thus, the method is applicable to a clinical
setting as patient specific parameters are able to be found
accurately and with minimal computation. This approach
ensures medical staff can have rapid patient specific
information to assist in diagnosis and therapy selection in
clinical real time (3-5 minutes).
V. CONCLUSION
Inflow
Fig. 2. Flows in and out for the model with optimized values versus the
model with the true values.
An integral based optimization method is presented which
turns a previously non-linear non-convex problem into a
linear convex problem. An example is given using a single
cardiac chamber to demonstrate the method. All parameters
were identified successfully and the results for rerunning the
model with the optimized parameters were very close to the
original simulated model. The differential equations are not
required to be solved, initial conditions are avoided and
there is a unique linear least squares solution to the
optimization equations.
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This method significantly reduces the computation
required an enables a fast and accurate method for
identifying patient specific parameters. The method readily
extends to the full minimal model and any non-linear
parameters can be replaced by a sufficient number of linear
parameters to ensure the optimization is linear and convex.
The integral approach allows the number of equations to be
greater than the number of parameters integrating over
different time intervals ensuring a unique solution.
Clinically, this approach means patient specific
parameters will be able to be found accurately and robustly
using a standard modern desktop computer. Medical staff
will be able to have rapid data on patients to assist in
diagnosis and can trial and test therapies in clinical real time
(3-5 minutes).
REFERENCES
[1] Smith B. W., Chase, J. G., Nokes, R. I., Shaw, G. M. and Wake,
G.. (2003). “Minimal haemodynamic system model including
ventricular interaction and valve dynamics.” J. Phys. Med. Eng.
Phys. Vol. , pp .
[2] Smith B. W. (2004). “Minimal Haemodynamic Modelling of the
Heart & Circulation for Clinical Application”, PhD thesis,
University of Canterbury.
[3] Carson, E. and Cobelli, C. (2001). “Modelling Methodology for
Physiology and Medicine.”
[4] Moore, C. L., Rose, G. A., Tayal, V. S., Sullivan, M. et al
(2002). “Determination of Left Ventricular Function by
Emergency Physician Echocardiography of Hypotensive
Patients.” Academic Emergency Medicine.